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Network N ℓ C Crand In this paper we outline some recent developments in the movie actors 225 226 3.65 0.79 0.00027 theory of social networks, particularly in the characteriza- neural network 282 2.65 0.28 0.05 tion and modeling of networks, and in the modeling of the power grid 4941 18.7 0.08 0.0005 spread of information or disease. Table 1: The number of nodes N, average degree of separa- tion ℓ, and clustering coefficient C, for three real-world net- 2 Random graphs works. The last column is the value which C would take in a with the same size and coordination num- The simplest explanation for the small-world effect uses the ber. idea of a random graph. Suppose there is some number N of people in the world, and on average they each have z 1 acquaintances. This means that there are 2 Nz connections A random graph does not show clustering. In a random between people in the entire population. The number z is graph the probability that two of person A’s friends will called the coordination number of the network. be friends of one another is no greater than the probabil- We can make a very simple model of a by ity that two randomly chosen people will be. On the other 1 taking N dots (“nodes” or “vertices”) and drawing 2 Nz hand, clustering has been shown to exist in a number of lines (“edges”) between randomly chosen pairs to repre- real-world networks. One can define a clustering coeffi- sent these connections. Such a network is called a random cient C, which is the average fraction of pairs of neigh- graph (Bollob´as 1985). Random graphs have been studied bors of a node which are also neighbors of each other. In a extensively in the community, particularly by fully connected network, in which everyone knows every- Erd¨os and R´enyi (1959). It is easy to see that a random one else, C = 1; in a random graph C = z/N, which is graph shows the small-world effect. If a person A on such a very small for a large network. In real-world networks it graph has z neighbors, and each of A’s neighbors also has z has been found that, while C is significantly less than 1, it 2 neighbors, then A has about z second neighbors. Extend- is much greater than O(N −1). In Table 1, we show some 3 4 ing this argument A also has z third neighbors, z fourth values of C calculated by Watts and Strogatz (1998) for neighbors and so on. Most people have between a hundred three different networks: the network of collaborations be- 4 and a thousand acquaintances, so z is already between tween movie actors discussed previously, the neural net- 8 12 about 10 and 10 , which is comparable with the popu- work of the worm C. Elegans, and the Western Power Grid lation of the world. In general the number D of degrees of the . We also give the value Crand which of separation which we need to consider in order to reach the clustering coefficient would have on random graphs of all N people in the network (also called the diameter of the same size and coordinationnumber, and in each case the the graph) is given by setting zD = N, which implies that measured value is significantly higher than for the random D = log N/ log z. This logarithmic increase in the number graph, indicating that indeed the graph is clustered. of degrees of separation with the size of the network is typ- In the same table we also show the average distance ℓ ical of the small-world effect. Since log N increases only between pairs of nodes in each of these networks. This is slowly with N, it allows the number of degrees to be quite not the same as the diameter D of the network discussed small even in very large systems. above, which is the maximum distance between nodes, but As an example of this type of behavior, Al- it also scales at most logarithmically with number of nodes bert et al. (1999) studied the properties of the network on random graphs. This is easy to see, since the average of “hyperlinks” between documents on the World Wide distance is strictly less than or equal to the maximum dis- Web. They estimated that, despite the fact there were tance, and so ℓ cannot increase any faster than D. As the 8 N 8 10 documents on the Web at the time the study table shows, the value of in each of the networks con- ≃ × ℓ was carried out, the average distance between documents sidered is small, indicating that the small-world effect is at was only about 19. work. (The precise definition of “small-world effect” is still There is a significant problemwith the randomgraph as a a matter of debate, but in the present case a reasonable defi- model of social networkshowever. The problem is that peo- nition would be that ℓ should be comparable with the value ple’s circles of acquaintance tend to overlap to a great ex- it would have on the random graph, which for the systems tent. Your friend’s friends are likely also to be your friends, discussed here it is.) or to put it another way, two of your friends are likely also So, if random graphs do not match well the properties to be friends with one another. This means that in a real of real-world networks, is there an alternative model which 2 social network it is not true to say that person A has z sec- does? Such a model has been suggested by Duncan Watts ond neighbors, since many of those friends of friends are and . It is described in the next section. also themselves friends of person A. This property is called clustering of networks.

2 (a) (b) (c)

Figure 1: (a) A one-dimensional lattice with each site connected to its z nearest neighbors, where in this case z =6. (b)The same lattice with periodic boundary conditions, so that the system becomes a ring. (c) The Watts–Strogatz model is created by rewiring a small fraction of the links (in this case five of them) to new sites chosen at random.

3 The small-world model of Watts which increase only slowly with system size. It is straight- and Strogatz forward to show that for a regular lattice in d dimensions which has the shape of a square or (hyper)cube of side L, d In order to model the real-world networks described in the and therefore has N = L vertices, the average vertex– 1/d last section, we need to find a way of generating graphs vertex distance increases as L, or equivalently as N . For which have both the clustering and small-world properties. small values of d this does not give us small-world behav- As we have argued, random graphs show the small-world ior. In one dimensionfor example, it means that the average effect, possessing average vertex-to-vertex distances which distance increases linearly with system size. If we allow the 1/d increase only logarithmically with the total number N of dimension d of the lattice to become large, then N be- vertices, but they do not show clustering—the property that comes a slowly increasing function of N, and so the lattice two neighbors of a vertex will often also be neighbors of does show the small-world effect. Could this be the expla- one another. nation for what we see in real networks? Perhaps real net- The opposite of a randomgraph, in some sense, is a com- works are roughly regular lattices of very high dimension. pletely ordered lattice, the simplest example of which is This explanation is in fact not unreasonable, although it has a one-dimensional lattice—a set of vertices arranged in a not been widely discussed. It works quitewell, providedthe straight line. If we take such a lattice and connect each ver- mean coordination number z of the vertices is much higher tex to the z vertices closest to it, as in Fig. 1a, then it is than twice the dimension d of the lattice. (If we allow z to easy to see that most of the immediate neighborsof any site approach 2d, then the clustering coefficient, Eq. (2), tends are also neighbors of one another, i.e., it shows the cluster- to zero, implying that the lattice loses its clustering proper- ing property. Normally, we apply periodic boundary con- ties.) ditions to the lattice, so that it wraps around on itself in Watts and Strogatz (1998) however have proposed an al- a ring (Fig. 1b), although this is just for convenience and ternative model for the small world, which perhaps fits bet- not strictly necessary. For such a lattice we can calculate ter with our everyday intuitions about the nature of social 2 networks. Their suggestion was to build a model which is, the clustering coefficient C exactly. As long as z < 3 N, which it will be for almost all graphs, we find that in essence, a low-dimensional regular lattice—say a one- dimensional lattice—but which has some degree of ran- 3(z 2) C = − , (1) domness in it, like a random graph, to produce the small- 4(z 1) world effect. They suggested a specific scheme for doing − 3 this as follows. We take the one-dimensional lattice of which tends to 4 in the limit of large z. We can also build networks out of higher-dimensional lattices, such as square Fig. 1b, and we go through each of the links on the lattice or cubic lattices, and these also show the clustering prop- in turn and, with some probability p, we randomly “rewire” erty. The value of the clustering coefficient in general di- that link, meaning that we move one of its ends to a new mension d is position chosen at random from the rest of the lattice. For 3(z 2d) small p this produces a graph with is still mostly regular but C = − , (2) 4(z d) has a few connections which stretch long distances across 3 − the lattice as in Fig. 1c. The coordination number of the which also tends to 4 for z 2d. Low-dimensional regular≫ lattices however do not show lattice is still z on average as it was before, although the the small-world effect of typical vertex–vertex distances number of neighbors of any particular vertex can be greater

3 or smaller than z. the distance itself, but this approach does not seem to have In social terms, we can justify this model by saying been tried for the Watts–Strogatz model.) In fact, it is pos- that, while most people are friends with their immediate sible to show that the series expansion of ℓ/L in powers of neighbors—neighbors on the same street, people that they p about p = 0 is well-behaved up to order pz−1, but that work with, people that their friends introduce them to— the expansion coefficients are infinite for all higher orders. some people are also friends with one or two people who For the version of the model where no links are ever re- are a long way away, in some social sense—people in other moved, the expansion coefficients take the same values up countries, people from other walks of life, acquaintances to order pz−1, but are finite for all higher orders as well. from previous eras of their lives, and so forth. These long- Generically, both versions of the model are referred to as distance acquaintances are represented by the long-range small-world models, or sometimes small-world graphs. links in the model of Watts and Strogatz. Many results have been derived for small-world models, Clearly the values of the clustering coefficient C for the and many of their other properties have been explored nu- Watts–Strogatz model with small values of p will be close merically. Here we give only a brief summary of the most to those for the perfectly ordered lattice given above, which important results. Barth´el´emy and Amaral (1999) conjec- 3 tend to 4 for fixed small d and large z. Watts and Stro- tured that the average vertex–vertex distance ℓ obeys the gatz also showed by numerical simulation that the average scaling form ℓ = ξG(L/ξ), where G(x) is a universal vertex–vertex distance ℓ is comparable with that for a true scaling function of its argument x and ξ is a characteris- random graph, even for quite small values of p. For ex- tic length-scale for the model which is assumed to diverge ample, for a random graph with N = 1000 and z = 10, in the limit of small p according to ξ p−τ . On the basis they found that the average distance was about ℓ =3.2 be- of numerical results, Barth´el´emy and∼ Amaral further con- 2 tween two vertices chosen at random. For their rewiring jectured that τ = 3 . Barrat (1999) disproved this second model, the average distance was only slightly greater, at conjecture using a simple physical argument which showed 1 ℓ = 3.6, when the rewiring probability p = 4 , compared that τ cannot be less than 1, and suggested on the basis of with ℓ = 50 for the graph with no rewired links at all. And more numerical results that in fact it was exactly 1. New- 1 even for p = 64 = 0.0156, they found ℓ = 7.4, a little man and Watts (1999b) showed that the small-world model over twice the value for the random graph. Thus the model has only one non-trivial length-scale other than the lattice appears to show both the clustering and small-world prop- spacing, which we can equate with the variable ξ above, erties simultaneously. This result has since been confirmed and which is given by by further simulation as well as analytic work on small- 1 world models, which is described in the next section. ξ = (3) pz for the one-dimensional model, or 4 Analytic and numerical results for 1 ξ = (4) small-world models (pzd)1/d

Most of the recent work on models of the small world in the general case. Thus τ must indeed be 1 for d = 1, or has been performed using a variation of the Watts–Strogatz τ =1/d for general d and, since there are no other length- model suggested by Newman and Watts (1999a). In this scales present, ℓ must be of the form version of the model, instead of rewiring links between sites L as in Fig. 1c, extra links, often called shortcuts, are added ℓ = F (pzLd), (5) 2dz between pairs of sites chosen at random, but no links are re- moved from the underlyinglattice. This model is somewhat where F (x) is another universal scaling function. (The ini- easier to analyze than the original Watts–Strogatz model, tial factor of (2d)−1 before the scaling function is arbitrary. because it is not possible for any region of the graph to be- It is chosen thus to give F a simple limit for small values come disconnected from the rest, whereas this can happen of its argument—see Eq. (6).) This scaling form is equiv- in the original model. Mathematically a disjoint section of alent to that of Barth´el´emy and Amaral by the substitution the graph can be represented by saying that the distance G(x) = xF (x) if τ = 1. It has been extensively con- from any vertex in that section to a vertex somewhere on firmed by numerical simulation (Newman and Watts 1999a, the rest of the graph is infinite. However, this means that, de Menezes et al. 2000) and by series expansions (New- when averaged over all possible realizations of the graph, man and Watts 1999b) (see Fig. 2). The divergence of ξ the average vertex–vertex distance ℓ in the model is also in- as p 0 gives something akin to a critical point in this finite for any finite value of p. (A similar problem in the limit.→ (De Menezes et al. (2000) have argued that, for tech- theory of random graphs is commonly dealt with by aver- nical reasons, we should refer to this point as a “first or- aging the reciprocal of vertex–vertex distance, rather than der critical point” (Fisher and Berker 1982).) This allowed

4 the average vertex-vertex distance by a factor of two, and 1.0 1.0 56 to reduce it by a factor of ten. L / n 0.5 In the limit of large p the small-world model becomes a 0.8 random graph or nearly so. Hence, we expect that the value

0.0 of ℓ should scale logarithmically with system size L when 0.6 0 4 8 p is large, and also, as the scaling form shows, when L is L / time t lz large. On the other hand, when p or L is small we expect 0.4 ℓ to scale linearly with L. This implies that F (x) has the limiting forms

0.2 1 for x 1 F (x)= ≪ (6)  (log x)/x for x 1. ≫ 0.0 0.1 1 10 100 1000 In theory there should be a leading constant in front of the pzL large-x form here, but, as discussed shortly, it turns out that this constant is equal to unity. The cross-over between the small- and large-x regimes must happen in the vicinity of Figure 2: Scaling collapse of average vertex–vertex dis- L = ξ, since ξ is the only length-scale available to dictate tances on d = 1 small-world graphs according to Eq. (5). this point. Points are numerical data for z = 2 (circles) and z = 10 Neither the actual distribution of path lengths in the (squares), for a variety of values of p and L. The solid line small-world model nor the average path length ℓ has been is a Pad´eapproximant derived from series expansions of calculated exactly yet; exact analytical calculations have the scaling function, while the dotted line is the mean-field proven very difficult for the model. Some exact results have solution, Eq. (8). Inset: the number of people infected as been given by Kulkarni et al. (2000) who show, for exam- a function of time by a disease which starts with a single ple, that the value of ℓ is simply related to the mean s and person and spreads through a community with the topology mean square s2 of the shortest distance s betweenh i two of a small-world graph. After Newman and Watts (1999b) points on diametricallyh i opposite sides of the graph, accord- and Newman et al. (2000). ing to ℓ s s2 = h i h i . (7) L L 1 − L(L 1) Newman and Watts (1999a) to apply a real-space renormal- − − ization group transformation to the model in the vicinity of Unfortunately, calculating the shortest distance between this point and prove that the scaling form above is exactly opposite points is just as difficult as calculating ℓ directly, obeyed in the limit of small p and large L. either analytically or numerically. Eq. (5) tells us that although the average vertex–vertex Newman et al. (2000) have calculated the form of the distance on a small-world graph appears at first glance to scaling function F (x) for d = 1 small-world graphs using be a function of three parameters—p, z, and L—it is in a mean-field-like approximation, which is exact for small fact entirely determined by a single scalar function of a sin- or large values of x, but not in the regime where x 1. ≃ gle scalar variable. If we know the form of this one func- Their result is tion, then we know everything. Actually, this statement is 4 −1 x strictly only true if ξ 1, when it is safe to ignoretheother F (x)= tanh . (8) √x2 +4x √x2 +4x length-scale in the problem,≫ the lattice parameter of the un- derlying lattice. Thus, the scaling form is expected to hold This form is also plotted on Fig. 2 (dotted line). Since this only when p is small, i.e., in the regime where the majority is exact for large x, it can be expanded about 1/x = 0 to of a person’s contacts are local and only a small fraction show that the leading constant in the large-x form of F (x), long-range. (The fourth parameter d also enters the equa- Eq. (6), is 1 as stated above. tion, but is not on an equal footingwith the others, since the Newman et al. also solved for the complete distribu- functional form of F changes with d, and thus Eq. (5) does tion of lengths between vertices in the model within their not tell us how ℓ varies with dimension.) mean-field approximation. This distribution can be used Both the scaling function F (x) and the scaling variable to give a simple model of the spread of a disease in a x pzLd have simple physical interpretations. The vari- small world. If a disease starts with a single person some- able≡ x is two times the average number of shortcuts on the where in the world, and spreads first to all the neighbors graph for the given value of p, and F (x) is the average frac- of that person, and then to all second neighbors, and so on, tion by which the vertex–vertex distance on the graph is re- then the number of people n who have the disease after duced for the given value of x. From the results shown in t time-steps is simply the number of people who are sep- 1 Fig. 2, we can see that it takes about 5 2 shortcuts to reduce arated from the initial carrier by a distance of t or less.

5 Newman and Watts (1999b) previously gave an approx- dimensional. On the other hand, as soon as p is greater imate differential equation for n(t) on an infinite small- than zero, the effective dimension of the graph becomes world graph, which they solved for the one-dimensional greater than one, and increases with system size (Newman case; Moukarzel (1999) later solved it for the case of gen- and Watts 1999b). Thus for any finite p we would expect eral d. The mean-field treatment generalizes the solution to see a phase transition at some finite temperature in the for d = 1 to finite lattice sizes. (A similar mean-field re- large system limit. Barrat and Weigt confirmed both ana- sult has been given for a slightly different disease-spreading lytically and numerically that indeed this is the case. The model by Kleczkowski and Grenfell (1999).) The resulting Ising model is of course a highly idealized model, and its form for n(t) is shown in the inset of Fig. 2, and clearly solution in this context is, to a large extent, just an inter- has the right general sigmoidal shape for the spread of an esting exercise. However, the similar problem of a Potts epidemic. In fact, this form of n is typical also of the antiferromagnet on a general graph has real practical ap- standard logistic growth models of disease spread, which plications, e.g., in the solution of scheduling problems. Al- are mostly based on random graphs (Sattenspiel and Si- though this problem has not been solved on the small-world mon 1988, Kretschmar and Morris 1996). In the next sec- graph, Walsh (1999) has found results which indicate that it tion we consider some (slightly) more sophisticated models may be interesting from a computational complexity point of disease spreading on small-world graphs. of view; finding a ground state for a Potts antiferromagnet on a small-world graph may be significantly harder than finding one on either a regular lattice or a random graph. 5 Other models based on small- Newman and Watts (1999b)looked at the problem of dis- ease spread on small-world graphs. As a first step away world graphs from the very simple models of disease described in the last section, they considered a disease to which only a cer- A variety of authors have looked at dynamical systems de- tain fraction q of the population is susceptible; the disease fined on small-world graphs built using either the Watts– spreads neighbor to neighbor on a small-world graph, ex- Strogatz rewiring method or the alternative method de- cept that it only affects, and can be transmitted by, the sus- scribed in Section 4. We briefly describe a number of these ceptible individuals. In such a model, the disease can only studies in this section. spread within the connected cluster of susceptible individ- Watts and Strogatz (1998, Watts 1999) looked at cellular uals in which it first starts, which is small if q is small, automata, simple games, and networks of coupled oscilla- but becomes larger, and eventually infinite, as q increases. tors on small-world networks. For example, they found that The point at which it becomes infinite—the point at which it was much easier for a cellular automaton to perform the an epidemic takes place—is precisely the percolation point task known as density classification (Das et al. 1994) on a for site percolation with probability q on the small-world small-world graph than on a regular lattice; they found that graph. Newman and Watts gave an approximate calcula- in an iterated multi-player game of Prisoner’s Dilemma, co- tion of this epidemic point, which compares reasonably fa- operation arose less frequently on a small-world graph than vorably with their numerical simulations. Moore and New- on a regular lattice; and they found that the small-world man (2000a, 2000b) later gave an exact solution. topology helped oscillator networks to synchronize much Lago-Fern´andez et al. (2000) investigated the behavior more easily than in the regular lattice case. of a neural network of Hodgkin–Huxley neurons on a va- Monasson (1999) investigated the eigenspectrum of the riety of graphs, including regular lattices, random graphs, Laplacian operator on small-world graphs using a transfer and small-world graphs. They found that the presence of matrix method. This spectrum tells us for example what the a high degree of clustering in the network allowed the net- normal modes would be of a system of masses and springs work to establish coherent oscillation, while short average built with the topology of a small-world graph. Or, perhaps vertex–vertex distances allowed the network to produce fast more usefully, it can tell us how diffusive dynamics would responses to changes in external stimuli. The small-world occur on a small world graph; any initial state of a diffusive graph, which simultaneously possesses both of these prop- field can be decomposed into eigenvectors which each de- erties, was the only graph they investigated which showed cay independently and exponentially with a decay constant both coherence and fast response. related to the corresponding eigenvalue. Diffusive motion Kulkarni et al. (1999) studied numerically the behavior might provide a simple model for the spread of information of the Bak–Sneppen model of species coevolution (Bak and of some kind in a social network. Sneppen 1993) on small-world graphs. This is a model Barrat and Weigt (2000) have given a solution of the fer- which mimics the evolutionary effects of interactions be- romagnetic Ising model on a d = 1 small-world network tween large numbers of species. The behavior of the model using a replica method. Since the Ising model has a lower is known to depend on the topology of the lattice on which critical dimension of two, we would expect it not to show it is situated, and Kulkarni and co-workers suggested that a phase transition when p = 0 and the graph is truly one-

6 ber of extra vertices in the middle which are connected to a large number of sites on the main lattice, chosen at random. (Lois Weisberg would be one of these extra sites.) This model is similar to the Watts–Strogatz model in that the ad- dition of the extra sites effectively introduces shortcuts be- tween randomly chosen positions on the lattice, so it should not be surprising to learn that this model does display the small-world effect. In fact, even in the case where only one extra site is added, the model shows the small-world effect if that site is sufficiently highly connected. This case has been solved exactly by Dorogovtsev and Mendes (1999). Another alternative model of the small world has been suggested by Albert et al. (1999)who, in their studies of the World Wide Web discussed in Section 2, concluded that the Web is dominated by a small number of very highly con- nected sites, as described above, but they also found that Figure 3: An alternative model of a small world, in which the distribution of the coordination numbers of sites (the there are a small number of individuals who are connected number of “hyperlinks” pointing to or from a site) was a to many widely-distributed acquaintances. power-law, rather than being bimodal as it is in the previ- ous model. They produced a model network of this kind as the topology of the small-world graph might be closer follows. Starting with a normal random graph with average to that of interactions in real ecosystems than the low- coordination number z and the desired number N of ver- dimensional regular lattices on which the Bak–Sneppen tices, they selected a vertex at random and added a link be- model is usually studied. The principal result of the sim- tween it and another randomly chosen site if that addition ulations was that on a small-world graph the amount of would bring the overall distribution of coordination num- evolutionary activity taking place at any given vertex varies bers closer to the required power law. Otherwise the vertex with the coordination number of the vertex, with the most remains as it is. If this process is repeated for a sufficiently connected nodes showing the greatest activity and the least long time, a network is generated with the correct coordi- connected ones showing the smallest. nation numbers, but which is in other respects a random graph. In particular, it does not show the clustering prop- erty of which such a fuss has been made in the case of the 6 Other models of the small world Watts–Strogatz model. Albert et al. found that their model matched the measured properties of the World Wide Web Although most of the work reviewed in this article is based quite closely, although related work by Adamic (1999)indi- on the Watts–Strogatz small-world model, a number of cates that clustering is present in the Web, so that the model other models of social networks have been proposed. In is unrealistic in this respect. Section 2 we mentioned the simple random-graph model It is worth noting that networks identical to those of Al- and in Section 3 we discussed a model based on a regular bert et al. can be generated in a manner much more effi- lattice of high dimension. In this section we describe briefly cient than the Monte Carlo scheme described above by sim- three others which have been suggested. ply generating N vertices with a power law distribution of One alternative to the view put forward by Watts and lines emerging from them (using, for instance, the trans- Strogatz is that the small-world phenomenon arises not be- formation method (Newman and Barkema 1999)), and then cause there are a few “long-range”connectionsin the other- joining pairs of lines together at random until none are left. wise short-range structure of a social network, but because If one were interested in investigating such networks nu- there are a few nodes in the network which have unusually merically, this would probably be the best way to generate high coordination numbers (Kasturirangan 1999) or which them. are linked to a widely distributed set of neighbors. Perhaps A third suggestion has been put forward by Klein- the “six degrees of separation” effect is due to a few people berg (1999), who argues that a model such as that of Watts who are particularly well connected. (Gladwell (1998) has and Strogatz, in which shortcuts connect vertices arbitrar- written a lengthy and amusing article arguing that a septua- ily far apart with uniform probability, is a poor represen- genarian salon proprietor in Chicago named Lois Weisberg tation of at least some real-world situations. (Kasturiran- is an example of precisely such a person.) A simple model gan (1999) has made a similar point.) Kleinberg notes that of this kind of network is depicted in Fig. 3, in which we in the real world, people are surprisingly good at finding start again with a one-dimensional lattice, but instead of short paths between pairs of individuals (Milgram’s letter adding extra links between pairs of sites, we add a num-

7 experiment, and the Kevin Bacon game are good exam- 2. This implies that information or disease spreading on ples) given only local information about the structure of a small-world graph reaches a number of people which the network. Conversely, he has shown that no algorithm increases initially as a power of time, then changes to exists which is capable of finding such paths on networks an exponential increase, and then flattens off as the of the Watts–Strogatz type, again given only local infor- graph becomes saturated. mation. Thus there must be some additional properties of 3. Disease models which incorporate a measure of sus- real-world networks which make it possible to find short ceptibility to infection have a percolation transition at paths with ease. To investigate this question further, Klein- which an epidemic sets in, whose position is influ- berg has proposed a generalization of the Watts–Strogatz enced strongly by the small-world nature of the net- model in which the typical distance traversed by the short- work. cuts can be tuned. Kleinberg’s model is based on a two- 4. Dynamical systems such as games or cellular au- dimensional square lattice (although it could be generalized tomata show quantitatively different behavior on to other dimensions d in a straightforward fashion) and has small-world graphs and regular lattices. Some prob- shortcuts added between pairs of vertices i, j with proba- − lems, such as density classification, appear to be easier bility which falls off as a power law d r of the distance be- ij to solve on small-world graphs, while others, such as tween them. (In this work, d is the “Manhattan distance” ij scheduling problems, appear to be harder. xi xj + yi yj , where (xi,yi) and (xj ,yj ) are the |lattice− coordinates| | − of| the vertices i and j. This makes good 5. Some real-world graphs show characteristics in addi- sense, since this is also the distance in terms of links on tion to the small-world effect which may be important the underlying lattice that separates those two points before to their function. An example is the World Wide Web, the shortcuts are added. However, one could in principle which appears to have a scale-free distribution of the generate networks using a different definition of distance, coordination numbers of vertices. such as the Euclidean distance (x x )2 + (y y )2, i − j i − j Research in this field is continuing in a variety of direc- for example.) It is then shown thatp for the particular value tions. Empirical work to determine the exact structure of r = 2 of the exponent of the power law (or r = d for real networks is underway in a number of groups, as well underlying lattices of d dimensions), there exists a simple as theoretical work to determine the properties of the pro- algorithm for finding a short path between two given ver- posed models. And studies to determine the effects of the tices, making use only of local information. For any other small-world topology on dynamical processes, although in value of r the problem of finding a short path is provably their infancy, promise an intriguing new perspective on the much harder. This result demonstrates that there is more way the world works. to the small world effect than simply the existence of short paths. Acknowledgements

7 Conclusions The author would like to thank Luis Amaral, Marc Barth´el´emy, Rahul Kulkarni, Cris Moore, Cristian In this article we have given an overview of recent theo- Moukarzel, Naomi Sachs, Steve Strogatz, Toby Walsh, and retical work on the “small-world” phenomenon. We have Duncan Watts for useful discussions and comments. This described in some detail the considerable body of recent work was supported in part by the and results dealing with the Watts–Strogatz small-world model DARPA under grant number ONR N00014-95-1-0975. and its variants, including analytic and numerical results about network structure and studies of dynamical systems on small-world graphs. References What have we learned from these efforts and where is this line of research going now? The most important ∗Citations of the form cond-mat/xxxxxxx refer to result is that small-world graphs—those possessing both the online condensed matter physics preprint archive at short average person-to-person distances and “clustering” http://www.arxiv.org/. of acquaintances—show behaviors very different from ei- ADAMIC, L. A. 1999 The small world web. Available ther regular lattices or random graphs. Some of the more as ftp://parcftp.xerox.com/pub/dynamics/- interesting such behaviors are the following: smallworld.ps. 1. These graphs show a transition with increasing num- ALBERT,R., JEONG, H. AND BARABASI´ , A.-L. 1999 Diam- ber of vertices from a “large-world” regime in which eter of the world-wide web. Nature 401, 130–131. the average distance between two people increases BAK, P. AND SNEPPEN, K. 1993 Punctuated equilibrium and linearly with system size, to a “small-world” one in criticality in a simple model of evolution. Physical Review which it increases logarithmically. Letters 71, 4083–4086.

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